Shahar Hod's bound on the fine-structure constant

[mitchell porter]

TL;DR Summary

does the fine structure constant saturate a quantum bound on the properties of charged black holes?

Shahar Hod is an Israeli physicist who has specialized in heuristic reasoning about the properties of quantum black holes, one might say in the tradition of Bekenstein and Hawking. By this I mean that, rather than trying to develop a detailed theory of quantum gravity, he has tried to obtain results through the use of general principles, e.g. the correspondence principle.

His biggest hit (over 800 citations) is a 1998 paper in which he argues that the areas of black holes (in natural units) are quantized in multiples of 4 hbar ln 3. For a while (Dreyer 2002, backstory here) there was excitement that loop quantum gravity could provide a detailed theory of quantum gravity in which this is true. That faded away, but Hod's work did lead to proofs (Motl 2002, Motl and Neitzke 2003) that Hod's factor of ln 3 appears in the frequencies of certain ringing modes of black holes ("quasinormal modes" that can lose energy e.g. via gravitational waves). To this degree, I believe, his 1998 work led to something that is now generally accepted.

Meanwhile, I have only now come across a 2010 work by Hod

"Gravitation, Thermodynamics, and the Fine-Structure Constant"

in which he adds a few extra principles and obtains a bound on the fine-structure constant that is extremely close to the measured value. The only comment on this argument that I can find, is a post by Lubos Motl. Lubos expresses his respect for Hod's work, says that his 2010 argument is similar in spirit to some of the hypotheses coming out of the "swampland" research program in quantum gravity, but that nonetheless he doesn't believe this particular argument.

Lubos offers two reasons for disagreeing. The most cogent is as follows. One of Hod's principles is that the minimum mass of a black hole should be one Planck mass. Lubos agrees that the lightest black holes should be of order the Planck mass, but the coefficient doesn't have to exactly equal 1, it's just going to have that order of magnitude, and that spoils the exactness of Hod's deduction.

His other reason is that he thinks there would be vacua in string theory, in which the fine-structure constant would violate Hod's bound, e.g. he considers alpha ~ 1/150 to be quite conceivable. But he doesn't go into any detail. I guess his reasoning is just "in the real world, the coupling starts at something like 1/24 at the GUT scale and runs down to 1/137 for electromagnetism, and I don't see why it couldn't have gone a bit further".

But the closeness of reality to Hod's bound - and the simplicity of his argument - is striking to me. I think it deserves a second look. And we are supposed to be in a new era of understanding the evaporation of near-extremal charged black holes; we had a thread about it just last month. If I had time, I would definitely be trying to evaluate Hod's argument from the perspective of JT gravity models of Reissner-Nordstrom black holes, which are the kind of black hole discussed in the thread, and also the kind of black hole which features in Hod's argument.

(P.S. For some reason I can't get the math markup right, if I get time I will fix it later.)

 

[ohwilleke]

one might say in the tradition of Bekenstein and Hawking.

As a complete aside, both of these men are now deceased, having died long before the life expectancy they had at birth. Many key mentors of both men survived them.

 

[mitchell porter]

I've studied a bit more of Hod's argument. He's looking at RN black holes, so charged, non-rotating black holes which have an inner and outer horizon that coincide when the mass radius equals the charge radius.

If the black hole emits a quantum with specific charge and mass, the black hole charge and mass will change, and so will the RN geometry. In particular we care about the distance between inner and outer horizons. They coincide if the mass radius and charge radius are the same. This actually means that the event horizon goes away, leaving just a naked singularity. This is often considered something to avoid, and so people seek a mechanism to enforce what Penrose called "cosmic censorship", according to which singularities are always hidden behind event horizons.

Hod instead focuses on the Hawking temperature of the black hole, which goes to zero when the horizons coincide. Hod says it would violate the third law of thermodynamics if the temperature can go to absolute zero in finite time, therefore Hawking radiation that would merge the two horizons must be forbidden. So in the end it's just another motivation for cosmic censorship (the classical motivation was that determinism fails at the singularity).

Hod has what I think is a formula for the de Broglie wave of a charged quantum of Hawking radiation in a given RN geometry. When the black hole emits a quantum of specific charge and mass, the black hole's own charge and mass change, and so does the geometry. So we're looking at a model of Hawking radiation and its consequences, similar to the "old quantum theory" of the hydrogen atom: stochastic transitions between Bohr orbits.

Hod restricts himself to RN black holes with integer charge - you could say that the corresponding RN geometries are his Bohr orbits - and then his resonance formula for the charged field describes a possible instance of Hawking radiation. We are to assume that Hawking radiation which would remove the event horizon is forbidden, and then we deduce the consequences.

In what follows, the number $ln(3)/(4\pi)$ (derived from a coefficient in his resonance formula) is important. First, its reciprocal provides a lower bound on the charge of the black hole: 12 times the charge of the electron (or proton, for positively charged black holes). Then, his lower bound on the fine structure constant is 1/12 times this number, i.e. $ln(3)/(48\pi)$. And as he observes, that's less than 0.2% below the actual value - so it seems that, for whatever reason, our world is close to saturating the bound.

I find it remarkable that this paper has provoked no online commentary, except for a Stack Exchange remark by Lubos Motl. If we take it seriously, we'd need to ask why is $\alpha$ so close to the bound? Hod's argument is just that there is a bound, not that the bound is saturated. Probably one could fashion an anthropic argument, but to me it suggests that the fine structure constant is dynamical, with some mechanism placing it at the edge of cosmic censorship violation; reminiscent of the criticality of the Higgs boson mass.